Sen.
Ted Cruz
of Texas raised about $4 million in the eight days after he formally announced his candidacy for the Republican presidential nomination, his campaign staff said, indicating donations are coming at a faster clip than for some GOP candidates in past years.

The fundraising tally comes as Mr. Cruz tries to answer questions about whether he can raise enough money to compete with big-donor magnets such as former Florida Gov.
Jeb Bush.

The donations show Mr. Cruz outraising in one week the $2.1 million garnered in two months by
Newt Gingrich,
a former House speaker, after formally launching his campaign in 2011.

Sen.
John McCain
(R., Ariz.) raised $1.7 million in the 1 ½ months after he started a presidential exploratory committee in 2006. He went on to win the party’s nomination that cycle.

“Often you have an establishment candidate, usually the moderate, who will be well funded,” said Cruz spokesman
Rick Tyler.
“Here we have a candidate who is conservative and can raise money.”

On March 23, Mr. Cruz became the first major candidate to formally launch a 2016 presidential bid.

His campaign is likely to rely more heavily on small-dollar donors, many identifying with the tea-party movement, than on the party’s wealthiest supporters. Of the total raised in the eight days after Mr. Cruz’s announcement, 95% were contributions of $100 or less, Mr. Tyler said. The average donation was about $83.

The Cruz campaign aims to raise about $40 million during the next year—a figure that could be dwarfed by the war chest being amassed by Mr. Bush, a favorite of the big-dollar GOP donors who is holding a series of nationwide fundraisers.

Mr. Cruz seems to be benefiting from the burst of attention he gained by his early entry into the race. A CBS News poll taken around the time of his announcement found 37% of Republicans would consider voting for the senator, up from 23% in February.

Mr. Cruz is campaigning this week in Iowa and South Carolina, two states that will vote early in the 2016 nominating contest.

Sen.
Rand Paul
of Kentucky, one of Mr. Cruz’s rivals for support among GOP conservatives, is set to formally announce his presidential bid on Tuesday.